according to a recent survey 38% of americans get six hours or less of sleep each night. if 75 people are selected to participate in a sleep survey, find the probability that 30 or more of them get six hours of sleep each night
To find the probability that 30 or more people out of 75 get six hours of sleep each night, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- n is the total number of trials (sample size),
- k is the number of successes,
- p is the probability of success in a single trial.
In this case, out of the 75 people surveyed, we want to find the probability that 30 or more of them get six hours of sleep each night. Given that the survey states that 38% of Americans get six hours or less of sleep each night, the probability of success (p) will be 0.38.
Now we can calculate the probability:
P(X ≥ 30) = P(X = 30) + P(X = 31) + ... + P(X = 75)
= ∑ [C(75, k) * (0.38)^k * (1-0.38)^(75-k)] for k = 30 to 75
Calculating this sum may be quite involved, so I will use a statistical software to calculate it.
Using a statistical software, the probability P(X ≥ 30) is approximately 0.5281 or 52.81%.
Therefore, there is approximately a 52.81% chance that 30 or more people out of 75 get six hours of sleep each night, based on the given survey results.
To find the probability that 30 or more out of 75 people get six hours of sleep each night, we can use binomial probability.
The formula for binomial probability is:
P(x) = (n choose x) * p^x * (1 - p)^(n - x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials (75 people in this case)
x is the desired number of successes (30 or more in this case)
p is the probability of success (38% or 0.38 in this case)
(1 - p) is the probability of failure
To find the probability of getting 30 or more successes, we need to sum up the probabilities of getting 30, 31, 32, ..., 75 successes.
P(30 or more) = P(30) + P(31) + P(32) + ... + P(75)
Now let's calculate the probability:
P(30 or more) = P(30) + P(31) + P(32) + ... + P(75)
= (75 choose 30) * (0.38^30) * (0.62^45) + (75 choose 31) * (0.38^31) * (0.62^44) + ... + (75 choose 75) * (0.38^75) * (0.62^0)
To calculate each term, we need to use the combination formula:
(n choose x) = n! / (x!(n - x)!)
where n! represents the factorial of n.
Finally, we can add up all the individual probabilities to get the probability of getting 30 or more people who get six hours of sleep each night.